Line a standard 12-cup muffin pan with paper liners and preheat the oven to 325°F (163°C).
In a medium bowl, combine the cookie or graham cracker crumbs, 3 tablespoons sugar, 5 tablespoons melted butter, and 1/8 teaspoon salt. Stir until the mixture resembles damp sand and holds together when pressed.
Divide the crumb mixture evenly among the 12 liners (about 1 to 1 1/2 tablespoons per cup). Press firmly into an even layer using the bottom of a small glass or measuring cup. Bake the crusts for 5 minutes, then set aside to cool slightly.
While the crusts bake, make the blackberry lavender sauce. In a small saucepan, combine blackberries, 2 tablespoons sugar, lemon juice, and lavender. Cook over medium heat, stirring and gently mashing the berries, until they release their juices and become jammy, 5 to 7 minutes.
In a small bowl, stir together cornstarch and 2 teaspoons cold water until smooth. Stir this slurry into the blackberry mixture and cook for 1 to 2 minutes more, until thickened and glossy. Remove from heat and let cool to lukewarm. If desired, press through a fine mesh sieve to remove seeds, then set aside.
For the cheesecake filling, place the softened cream cheese in a mixing bowl. Beat on medium speed until completely smooth and creamy, scraping down the bowl as needed.
Add 1/2 cup sugar, the remaining 1/8 teaspoon salt, and the sour cream. Beat on medium-low until smooth and well combined, avoiding incorporating too much air.
Add the eggs one at a time, mixing on low just until each egg is incorporated before adding the next. Scrape down the bowl as needed.
Add the heavy cream and vanilla extract, and mix on low until the batter is smooth and uniform. Do not overbeat.
Divide the cheesecake batter evenly among the prepared crusts, filling each liner almost to the top.
Spoon small dollops (about 1/2 teaspoon each) of the cooled blackberry lavender sauce over the surface of each cheesecake, using 1 to 2 teaspoons sauce per cheesecake. Use a toothpick or skewer to gently swirl the sauce into the top of the batter without digging into the crust.
Bake the mini cheesecakes at 325°F for 16 to 20 minutes, or until the edges are set and the centers still have a slight wobble when the pan is gently nudged.
Turn off the oven, crack the door open slightly, and let the cheesecakes sit in the oven for 10 minutes to begin cooling gradually.
Remove the pan from the oven and let the cheesecakes cool to room temperature in the pan, about 45 to 60 minutes.
Once cooled, transfer the pan to the refrigerator and chill the cheesecakes for at least 1 hour, or until fully cold and set.
Carefully remove the mini cheesecakes from the muffin pan. Peel away the liners just before serving. Garnish with a fresh blackberry or a thin lemon slice if desired.